Birational Darboux Coordinates on (Co)Adjoint Orbits of GL(N, ℂ)

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4 Scopus citations


The set of all linear transformations with a fixed Jordan structure J is a symplectic manifold isomorphic to the coadjoint orbit O(J) of the general linear group GL(N, C). Any linear transformation can be projected along its eigenspace onto a coordinate subspace of complementary dimension. The Jordan structure (Formula presented.) of the image under the projection is determined by the Jordan structure J of the preimage; consequently, the projection (Formula presented.) is a mapping of symplectic manifolds. It is proved that the fiber ℰ of the projection is a linear symplectic space and the map (Formula presented.) is a birational symplectomorphism. Successively projecting the resulting transformations along eigensubspaces yields an isomorphism between O(J) and the linear symplectic space being the direct product of all fibers of the projections. The Darboux coordinates on O(J) are pullbacks of the canonical coordinates on this linear symplectic space. Canonical coordinates on orbits corresponding to various Jordan structures are constructed as examples.

Original languageEnglish
Pages (from-to)17-30
Number of pages14
JournalFunctional Analysis and its Applications
Issue number1
StatePublished - 1 Jan 2016

Scopus subject areas

  • Analysis
  • Applied Mathematics


  • birational symplectic coordinates
  • Jordan normal form
  • Lie-Poisson-Kirillov-Kostant form

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